ASYMPTOTYC FORMULAE CONCERNING ARITHMETICAL FUNCTIONS DEFINED BY CROSS-CONVOLUTIONS , VII. DISTRIBUTION OF A-SEMI-k-FREE INTEGERS

Abstract

We define the A-semi-k-free integers as a common generalization of the k-free integers (i.e. integers not divisible by the k-th power of any prime) and of the semi-k-free integers (i.e. integers not divisible unitarily by the k-th power of any prime) in terms of Narkiewicz's regular A-convolutions. We establish asymptotic formulae for the number of A-semi-l-free integers £x with and without assuming the Riemann hypothesis if A is a cross-convolution, investigated in our previous papers.